Solve 199.667-170.333/sqrt{frac{8.505^2{3}+34.766^2/15}}

$$\frac{ 199.667-170.333 }{ \sqrt{ \frac{ { 8.505 }^{ 2 } }{ 3 } + \frac{ { 34.766 }^{ 2 } }{ 15 } } }$$

Evaluate

$$\frac{29334\sqrt{23555248215}}{1570349881}\approx 2.866940661$$

Solution Steps

Subtract 170.333 from 199.667 to get 29.334.

$$\frac{29.334}{\sqrt{\frac{8.505^{2}}{3}+\frac{34.766^{2}}{15}}}$$

Calculate 8.505 to the power of 2 and get 72.335025.

$$\frac{29.334}{\sqrt{\frac{72.335025}{3}+\frac{34.766^{2}}{15}}}$$

Expand \frac{72.335025}{3} by multiplying both numerator and the denominator by 1000000.

$$\frac{29.334}{\sqrt{\frac{72335025}{3000000}+\frac{34.766^{2}}{15}}}$$

Reduce the fraction \frac{72335025}{3000000} to lowest terms by extracting and canceling out 75.

$$\frac{29.334}{\sqrt{\frac{964467}{40000}+\frac{34.766^{2}}{15}}}$$

Calculate 34.766 to the power of 2 and get 1208.674756.

$$\frac{29.334}{\sqrt{\frac{964467}{40000}+\frac{1208.674756}{15}}}$$

Expand \frac{1208.674756}{15} by multiplying both numerator and the denominator by 1000000.

$$\frac{29.334}{\sqrt{\frac{964467}{40000}+\frac{1208674756}{15000000}}}$$

Reduce the fraction \frac{1208674756}{15000000} to lowest terms by extracting and canceling out 4.

$$\frac{29.334}{\sqrt{\frac{964467}{40000}+\frac{302168689}{3750000}}}$$

Least common multiple of 40000 and 3750000 is 15000000. Convert \frac{964467}{40000} and \frac{302168689}{3750000} to fractions with denominator 15000000.

$$\frac{29.334}{\sqrt{\frac{361675125}{15000000}+\frac{1208674756}{15000000}}}$$

Since \frac{361675125}{15000000} and \frac{1208674756}{15000000} have the same denominator, add them by adding their numerators.

$$\frac{29.334}{\sqrt{\frac{361675125+1208674756}{15000000}}}$$

Add 361675125 and 1208674756 to get 1570349881.

$$\frac{29.334}{\sqrt{\frac{1570349881}{15000000}}}$$

Rewrite the square root of the division \sqrt{\frac{1570349881}{15000000}} as the division of square roots \frac{\sqrt{1570349881}}{\sqrt{15000000}}.

$$\frac{29.334}{\frac{\sqrt{1570349881}}{\sqrt{15000000}}}$$

Factor 15000000=1000^{2}\times 15. Rewrite the square root of the product \sqrt{1000^{2}\times 15} as the product of square roots \sqrt{1000^{2}}\sqrt{15}. Take the square root of 1000^{2}.

$$\frac{29.334}{\frac{\sqrt{1570349881}}{1000\sqrt{15}}}$$

Rationalize the denominator of \frac{\sqrt{1570349881}}{1000\sqrt{15}} by multiplying numerator and denominator by \sqrt{15}.

$$\frac{29.334}{\frac{\sqrt{1570349881}\sqrt{15}}{1000\left(\sqrt{15}\right)^{2}}}$$

The square of \sqrt{15} is 15.

$$\frac{29.334}{\frac{\sqrt{1570349881}\sqrt{15}}{1000\times 15}}$$

To multiply \sqrt{1570349881} and \sqrt{15}, multiply the numbers under the square root.

$$\frac{29.334}{\frac{\sqrt{23555248215}}{1000\times 15}}$$

Multiply 1000 and 15 to get 15000.

$$\frac{29.334}{\frac{\sqrt{23555248215}}{15000}}$$

Divide 29.334 by \frac{\sqrt{23555248215}}{15000} by multiplying 29.334 by the reciprocal of \frac{\sqrt{23555248215}}{15000}.

$$\frac{29.334\times 15000}{\sqrt{23555248215}}$$

Rationalize the denominator of \frac{29.334\times 15000}{\sqrt{23555248215}} by multiplying numerator and denominator by \sqrt{23555248215}.

$$\frac{29.334\times 15000\sqrt{23555248215}}{\left(\sqrt{23555248215}\right)^{2}}$$

The square of \sqrt{23555248215} is 23555248215.

$$\frac{29.334\times 15000\sqrt{23555248215}}{23555248215}$$

Multiply 29.334 and 15000 to get 440010.

$$\frac{440010\sqrt{23555248215}}{23555248215}$$

Divide 440010\sqrt{23555248215} by 23555248215 to get \frac{29334}{1570349881}\sqrt{23555248215}.

$$\frac{29334}{1570349881}\sqrt{23555248215}$$